# How do you rationalize the denominator (sqrt27-sqrt7)/(sqrt21-3)?

Mar 24, 2018

$\frac{\sqrt{27} - \sqrt{7}}{\sqrt{21} - 3}$

Multiply both the numerator and denominator with $\left(\sqrt{21} + 3\right)$,

$\frac{\sqrt{27} - \sqrt{7}}{\sqrt{21} - 3} \times \frac{\sqrt{21} + 3}{\sqrt{21} + 3}$

Apply difference of two squares rule,

$\frac{\left(\sqrt{27} - \sqrt{7}\right) \left(\sqrt{21} + 3\right)}{{\sqrt{21}}^{2} - {3}^{2}}$

Expand,

$\frac{\left(\sqrt{27}\right) \left(\sqrt{21}\right) + \left(- \sqrt{7}\right) \left(\sqrt{21}\right) + \left(\sqrt{27}\right) \left(3\right) + \left(- \sqrt{7}\right) \left(3\right)}{21 - 9}$

Simplify,

$\frac{6 \sqrt{7} + 2 \sqrt{3}}{12}$

SImplest form,

$\frac{3 \sqrt{7} + 2 \sqrt{3}}{6}$